Condition REF  requires that the first and second-order derivatives of the density \(f_{(s)}(\cdot )\)  have a stronger continuous property so that there is enough information around the quasi-true parameter \(\beta ^\ast _{(s)}\) . When each \(x_{{(s)},r}\)  is i.i.d. \(N(0,1)\) , we have \(|f^{\prime }_{(s)}(t_1)-f^{\prime }_{(s)}(t_2)|\le (2\pi )^{-1/2}|t_1-t_2|\)  and \(|f^{\prime \prime }_{(s)}(t_1)-f^{\prime \prime }_{(s)}(t_2)|\le \sqrt{2/\pi }|t_1-t_2|\) .
Conditions REF  and REF  are also similarly used in Lemmas .2–.4 of [1]} and Condition (ii) of [2]}. Note that we require uniformity across \(s\)  in this condition because our averaging allows the number of candidate models to diverge.
